Integral de $$$\sec^{2}{\left(2 x \right)}$$$

Halla $$$\int \sec^{2}{\left(2 x \right)}\, dx$$$.

Sea $$$u=2 x$$$.

Entonces $$$du=\left(2 x\right)^{\prime }dx = 2 dx$$$ (los pasos pueden verse »), y obtenemos que $$$dx = \frac{du}{2}$$$.

La integral se convierte en

$${\color{red}{\int{\sec^{2}{\left(2 x \right)} d x}}} = {\color{red}{\int{\frac{\sec^{2}{\left(u \right)}}{2} d u}}}$$

Aplica la regla del factor constante $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ con $$$c=\frac{1}{2}$$$ y $$$f{\left(u \right)} = \sec^{2}{\left(u \right)}$$$:

$${\color{red}{\int{\frac{\sec^{2}{\left(u \right)}}{2} d u}}} = {\color{red}{\left(\frac{\int{\sec^{2}{\left(u \right)} d u}}{2}\right)}}$$

La integral de $$$\sec^{2}{\left(u \right)}$$$ es $$$\int{\sec^{2}{\left(u \right)} d u} = \tan{\left(u \right)}$$$:

$$\frac{{\color{red}{\int{\sec^{2}{\left(u \right)} d u}}}}{2} = \frac{{\color{red}{\tan{\left(u \right)}}}}{2}$$

Recordemos que $$$u=2 x$$$:

$$\frac{\tan{\left({\color{red}{u}} \right)}}{2} = \frac{\tan{\left({\color{red}{\left(2 x\right)}} \right)}}{2}$$

Por lo tanto,

$$\int{\sec^{2}{\left(2 x \right)} d x} = \frac{\tan{\left(2 x \right)}}{2}$$

Añade la constante de integración:

$$\int{\sec^{2}{\left(2 x \right)} d x} = \frac{\tan{\left(2 x \right)}}{2}+C$$

$$$\int \sec^{2}{\left(2 x \right)}\, dx = \frac{\tan{\left(2 x \right)}}{2} + C$$$A